The scheme requires extending the node values to all \(x\in {\mathbb {R}}^d\)  using an interpolation \(\mathbb {I}\) . We denote by \(\mathbb {I}\left[V_\Delta \right] (x,q)\)  the interpolation of the values \(V_\Delta (x_j,q)\)  computed at \((x,q)\) . With this notation, a standard semi-Lagrangian discretization of the Hamiltonian is given (see [1]}) by
\(\Sigma \left(x_j,q,V_\Delta \right)= \\h\ell (x_j) + e^{-\lambda h}\min _{u\in U}\left\lbrace \mathbb {I}\left[V_\Delta \right] (x_j+h \: f(x_j,q,u), q)\right\rbrace .\) 
